Independence proof concerning compact cardinals, continuum hypotbesis, and normal ultrafilters

关于紧基数、连续统假设和普通超滤器的独立性证明

• 批准号: 09640299
• 负责人: ABE Yoshihiro
• 金额: $ 0.9万
• 依托单位: Kanagawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640299/
• 关键词: compact cardinal; continuum hypothesis; normal ultrafilter; forcing; indescribability; bounded ideal; partition property; precipitous; Precipitous; compact cardinal; seminormal ideal; clused unbounded filter; diamond principle; very weak squa

We investigated a K<@D1+@>D1-C.C.(]SY.di-substituted right.[)K-strategically closed forcing adding arbitrary nunber of closed unbounded subsets of K.A Lottery preparation followed by this forcing preserves the strong compactness of K.In the resulting model the restricions of all normal ultrafilters to the ground model coincide.We know a symmetric model in which the genemlized continuum hypothesis holds below a strongly compact cardinal K and fails at K.However it turned out establishing the same thing together with the axiom of choice is far more difficult than had been expected. It is not clear whether the strong compactness of K is preserved when we force the the axion of choice on the symmetric model.Several facts and new technique are founded as in the folllowing.Forcing : (1)Adding many closed unbounded sets (2)Adding non-reflecting stationary sets to P_klambda (3)Forcing a non-regular ultrafilter on P_kK^<+++> with K supercompact.Combinatorics : (1)Combinatorial characterization of PIi-indescribability in P_klambda.(2)Bounded ideal may or may not have the partition property.(3)Bounded ideal is not precipitousIt is expected that further research makes the relation between the partition property and ineffability of P_klambda clear to bring great progress in the combinatorics of P_klambda.

我们研究了一个K<@D1+@>D1-C.C.（]SY.双取代右。[）K-在K的无界闭子集上增加任意个数的策略闭强迫.在此强迫下的Lottery准备保持了K的强紧性.在所得到的模型中，所有正规超滤子对基模型的约束一致.我们知道一个对称模型，其中一般的连续统假设在强紧基数K以下成立，在K处失败.然而，它证明了在一起建立同样的东西与选择公理的关系远比预期的要困难得多。在对称模型上强制选择轴子时，K的强紧性是否保持还不清楚.我们发现了如下事实和新技巧：强制：（1）增加许多闭无界集（2）在P_klambda上增加非反射平稳集（3）在P_kK^<+>上强制K超紧的非正则超滤子.（1）P_k λ中PI_i可解性的组合刻画。（2）有界理想可以有也可以没有分拆性质。（3）有界理想并不陡峭，希望通过进一步的研究，使P_k λ的分划性与不可描述性之间的关系更加清晰，从而使P_k λ的组合学得到更大的发展。

项目成果

Yoshihiro Abe: "A variant of the diamond principle for combinatorial ideals" Proceedings of the American Mathematical Society. 127-3. 847-849 (1999)

Yoshihiro Abe：“组合理想的钻石原理的变体”美国数学会论文集。

Sakae Fuchino and Lajos Soukup: "More set-theory around the Weak Freese-Nation property" Fundamenta Mathematicae. 154-2. 159-176 (1997)

Sakae Fuchino 和 Lajos Soukup：“有关弱自由国家性质的更多集合论”基础数学。

Yoshihiro Abe: "A variant of the diamond principle for combinatorial ideals" Proceedings of the American Mathematical society. (1998)

安倍佳弘：“组合理想的钻石原理的变体”美国数学会论文集。

Yo Matsubara and Masahiro Shioya: "Nowhere precipitousness of some ideals" Journal of Symbolic Logic. 63. 1003-1006 (1998)

Yo Matsubara 和 Masahiro Shioya：“某些理想的陡峭性”《符号逻辑杂志》。

Yo Matsubara and Masahiro Shioya: "Nowbere precipitousness of some ideals" Journal of Symbolic Logic. 63-2. 1003-1006 (1998)

Yo Matsubara 和 Masahiro Shioya：“现在有些理想的陡峭性”《符号逻辑杂志》。